Abstract
We present a microscopic model of black hole (BH) ‘evaporation’ in asymptotically AdS2 spacetimes dual to the low energy sector of the SYK model. To describe evaporation, the SYK model is coupled to a bath comprising of Nf free scalar fields Φi. We consider a linear combination of couplings of the form OSY K(t)∑iΦi(0, t), where OSY K involves products of the Kourkoulou-Maldacena operator \( iJ/N{\sum}_{k=1}^{N/2}{s}_k^{\prime }{\psi}_{2k-1}(t){\psi}_{2k}(t) \) specified by a spin vector s′. We discuss the time evolution of a product of (i) a pure state of the SYK system, namely a BH microstate characterized by a spin vector s and an effective BH temperature TBH, and (ii) a Calabrese-Cardy state of the bath characterized by an effective temperature Tbath. We take Tbath ≪ TBH, and TBH much lower than the characteristic UV scale J of the SYK model, allowing a description in terms of the time reparameterization mode. Tracing over the bath degrees of freedom leads to a Feynman-Vernon type effective action for the SYK model, which we study in the low energy limit. The leading large N behaviour of the time reparameterization mode is found, as well as the \( O\left(1/\sqrt{N}\right) \) fluctuations. The latter are characterized by a non-Markovian non-linear stochastic differential equation with non-local Gaussian noise. In a restricted range of couplings, we find two classes of solutions which asymptotically approach (a) a BH at a lower temperature, and (b) a horizonless geometry. We identify these with partial and complete BH evaporation, respectively. Importantly, the asymptotic solution in both cases involves the scalar product of the spin vectors s.s′, which carries some information about the initial state. By repeating the dynamical process O(N2) times with different choices of the spin vector s′, one can in principle reconstruct the initial BH microstate.
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References
S. Sachdev and J. Ye, Gapless spin fluid ground state in a random, quantum Heisenberg magnet, Phys. Rev. Lett. 70 (1993) 3339 [cond-mat/9212030] [INSPIRE].
A. Kitaev, A simple model of quantum holography, talks at KITP, 7 April and 27 May 2015.
J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 106002 [arXiv:1604.07818] [INSPIRE].
S. Sachdev, Bekenstein-Hawking Entropy and Strange Metals, Phys. Rev. X 5 (2015) 041025 [arXiv:1506.05111] [INSPIRE].
J. Maldacena, D. Stanford and Z. Yang, Conformal symmetry and its breaking in two dimensional Nearly Anti-de-Sitter space, PTEP 2016 (2016) 12C104 [arXiv:1606.01857] [INSPIRE].
G. Sárosi, AdS2 holography and the SYK model, PoS Modave2017 (2018) 001 [arXiv:1711.08482] [INSPIRE].
D.A. Trunin, Pedagogical introduction to the Sachdev–Ye–Kitaev model and two-dimensional dilaton gravity, Usp. Fiz. Nauk 191 (2021) 225 [arXiv:2002.12187] [INSPIRE].
D. Stanford and E. Witten, Fermionic Localization of the Schwarzian Theory, JHEP 10 (2017) 008 [arXiv:1703.04612] [INSPIRE].
L.V. Iliesiu and G.J. Turiaci, The statistical mechanics of near-extremal black holes, JHEP 05 (2021) 145 [arXiv:2003.02860] [INSPIRE].
J.D. Bekenstein, Black holes and the second law, Lett. Nuovo Cim. 4 (1972) 737 [INSPIRE].
J.D. Bekenstein, Black holes and entropy, Phys. Rev. D 7 (1973) 2333 [INSPIRE].
S.W. Hawking, Black hole explosions, Nature 248 (1974) 30 [INSPIRE].
S.W. Hawking, Particle Creation by Black Holes, Commun. Math. Phys. 43 (1975) 199 [Erratum ibid. 46 (1976) 206] [INSPIRE].
A. Strominger and C. Vafa, Microscopic origin of the Bekenstein-Hawking entropy, Phys. Lett. B 379 (1996) 99 [hep-th/9601029] [INSPIRE].
J.R. David, G. Mandal and S.R. Wadia, Microscopic formulation of black holes in string theory, Phys. Rept. 369 (2002) 549 [hep-th/0203048] [INSPIRE].
J.M. Maldacena, The Large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [hep-th/9711200] [INSPIRE].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
E. Witten, Anti-de Sitter space, thermal phase transition, and confinement in gauge theories, Adv. Theor. Math. Phys. 2 (1998) 505 [hep-th/9803131] [INSPIRE].
G. Penington, Entanglement Wedge Reconstruction and the Information Paradox, JHEP 09 (2020) 002 [arXiv:1905.08255] [INSPIRE].
A. Almheiri, N. Engelhardt, D. Marolf and H. Maxfield, The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole, JHEP 12 (2019) 063 [arXiv:1905.08762] [INSPIRE].
A. Almheiri, R. Mahajan, J. Maldacena and Y. Zhao, The Page curve of Hawking radiation from semiclassical geometry, JHEP 03 (2020) 149 [arXiv:1908.10996] [INSPIRE].
A. Almheiri, R. Mahajan and J. Maldacena, Islands outside the horizon, arXiv:1910.11077 [INSPIRE].
M. Rozali et al., Information radiation in BCFT models of black holes, JHEP 05 (2020) 004 [arXiv:1910.12836] [INSPIRE].
G. Penington, S.H. Shenker, D. Stanford and Z. Yang, Replica wormholes and the black hole interior, JHEP 03 (2022) 205 [arXiv:1911.11977] [INSPIRE].
A. Almheiri et al., Replica Wormholes and the Entropy of Hawking Radiation, JHEP 05 (2020) 013 [arXiv:1911.12333] [INSPIRE].
H.Z. Chen et al., Evaporating Black Holes Coupled to a Thermal Bath, JHEP 01 (2021) 065 [arXiv:2007.11658] [INSPIRE].
A. Almheiri et al., The entropy of Hawking radiation, Rev. Mod. Phys. 93 (2021) 035002 [arXiv:2006.06872] [INSPIRE].
H. Geng and A. Karch, Massive islands, JHEP 09 (2020) 121 [arXiv:2006.02438] [INSPIRE].
H. Geng et al., Inconsistency of islands in theories with long-range gravity, JHEP 01 (2022) 182 [arXiv:2107.03390] [INSPIRE].
C. Krishnan, Critical Islands, JHEP 01 (2021) 179 [arXiv:2007.06551] [INSPIRE].
K. Ghosh and C. Krishnan, Dirichlet baths and the not-so-fine-grained Page curve, JHEP 08 (2021) 119 [arXiv:2103.17253] [INSPIRE].
C. Krishnan, V. Patil and J. Pereira, Page Curve and the Information Paradox in Flat Space, arXiv:2005.02993 [INSPIRE].
C. Krishnan and V. Mohan, Interpreting the Bulk Page Curve: A Vestige of Locality on Holographic Screens, arXiv:2112.13783 [INSPIRE].
S. Raju, Failure of the split property in gravity and the information paradox, Class. Quant. Grav. 39 (2022) 064002 [arXiv:2110.05470] [INSPIRE].
J. De Vuyst and T.G. Mertens, Operational islands and black hole dissipation in JT gravity, JHEP 01 (2023) 027 [arXiv:2207.03351] [INSPIRE].
E. Bahiru et al., State-dressed local operators in AdS/CFT, arXiv:2209.06845 [INSPIRE].
A. Almheiri, A. Milekhin and B. Swingle, Universal Constraints on Energy Flow and SYK Thermalization, arXiv:1912.04912 [INSPIRE].
J. Maldacena and A. Milekhin, SYK wormhole formation in real time, JHEP 04 (2021) 258 [arXiv:1912.03276] [INSPIRE].
Y. Chen, X.-L. Qi and P. Zhang, Replica wormhole and information retrieval in the SYK model coupled to Majorana chains, JHEP 06 (2020) 121 [arXiv:2003.13147] [INSPIRE].
C. Teitelboim, Gravitation and Hamiltonian Structure in Two Space-Time Dimensions, Phys. Lett. B 126 (1983) 41 [INSPIRE].
R. Jackiw, Lower Dimensional Gravity, Nucl. Phys. B 252 (1985) 343 [INSPIRE].
I. Kourkoulou and J. Maldacena, Pure states in the SYK model and nearly-AdS2 gravity, arXiv:1707.02325 [INSPIRE].
J. Engelsöy, T.G. Mertens and H. Verlinde, An investigation of AdS2 backreaction and holography, JHEP 07 (2016) 139 [arXiv:1606.03438] [INSPIRE].
P. Calabrese and J. Cardy, Quantum Quenches in Extended Systems, J. Stat. Mech. 0706 (2007) P06008 [arXiv:0704.1880] [INSPIRE].
J. Cardy, Thermalization and Revivals after a Quantum Quench in Conformal Field Theory, Phys. Rev. Lett. 112 (2014) 220401 [arXiv:1403.3040] [INSPIRE].
A. Dhar et al., Gravitational collapse in SYK models and Choptuik-like phenomenon, JHEP 11 (2019) 067 [arXiv:1812.03979] [INSPIRE].
K. Papadodimas and S. Raju, State-Dependent Bulk-Boundary Maps and Black Hole Complementarity, Phys. Rev. D 89 (2014) 086010 [arXiv:1310.6335] [INSPIRE].
K. Papadodimas and S. Raju, Remarks on the necessity and implications of state-dependence in the black hole interior, Phys. Rev. D 93 (2016) 084049 [arXiv:1503.08825] [INSPIRE].
A.R. Brown, H. Gharibyan, G. Penington and L. Susskind, The Python’s Lunch: geometric obstructions to decoding Hawking radiation, JHEP 08 (2020) 121 [arXiv:1912.00228] [INSPIRE].
G. Mandal and S.R. Wadia, Black hole geometry around an elementary BPS string state, Phys. Lett. B 372 (1996) 34 [hep-th/9511218] [INSPIRE].
A. Dhar, G. Mandal and S.R. Wadia, Absorption versus decay of black holes in string theory and T symmetry, Phys. Lett. B 388 (1996) 51 [hep-th/9605234] [INSPIRE].
P. Calabrese and J.L. Cardy, Evolution of entanglement entropy in one-dimensional systems, J. Stat. Mech. 0504 (2005) P04010 [cond-mat/0503393] [INSPIRE].
D. Bagrets, A. Altland and A. Kamenev, Sachdev–Ye–Kitaev model as Liouville quantum mechanics, Nucl. Phys. B 911 (2016) 191 [arXiv:1607.00694] [INSPIRE].
A. Almheiri, A. Mousatov and M. Shyani, Escaping the interiors of pure boundary-state black holes, JHEP 02 (2023) 024 [arXiv:1803.04434] [INSPIRE].
G. Mandal, R. Sinha and N. Sorokhaibam, Thermalization with chemical potentials, and higher spin black holes, JHEP 08 (2015) 013 [arXiv:1501.04580] [INSPIRE].
G. Mandal, S. Paranjape and N. Sorokhaibam, Thermalization in 2D critical quench and UV/IR mixing, JHEP 01 (2018) 027 [arXiv:1512.02187] [INSPIRE].
P. Banerjee, A. Gaikwad, A. Kaushal and G. Mandal, Quantum quench and thermalization to GGE in arbitrary dimensions and the odd-even effect, JHEP 09 (2020) 027 [arXiv:1910.02404] [INSPIRE].
J.V. Rocha, Evaporation of large black holes in AdS: Coupling to the evaporon, JHEP 08 (2008) 075 [arXiv:0804.0055] [INSPIRE].
A. Kamenev, Field theory of non-equilibrium systems, Cambridge University Press (2011).
F.M. Haehl, R. Loganayagam and M. Rangamani, Schwinger-Keldysh formalism. Part I: BRST symmetries and superspace, JHEP 06 (2017) 069 [arXiv:1610.01940] [INSPIRE].
R.F. Pawula, Generalizations and extensions of the Fokker-Planck-Kolmogorov equations, IEEE Trans. Inf. Theory 13 (1967) 33.
H. Risken and H.D. Vollmer, On the application of truncated generalized Fokker-Planck equations, Zeitschrift für Physik B Condensed Matter 35 (1979) 313.
J.D. Jackson, Classical Electrodynamics, Wiley (2012).
A.O. Caldeira and A.J. Leggett, Path integral approach to quantum Brownian motion, Physica A 121 (1983) 587 [INSPIRE].
P. Gao, D.L. Jafferis and A.C. Wall, Traversable Wormholes via a Double Trace Deformation, JHEP 12 (2017) 151 [arXiv:1608.05687] [INSPIRE].
J. Maldacena, D. Stanford and Z. Yang, Diving into traversable wormholes, Fortsch. Phys. 65 (2017) 1700034 [arXiv:1704.05333] [INSPIRE].
J. Maldacena and X.-L. Qi, Eternal traversable wormhole, arXiv:1804.00491 [INSPIRE].
R.K. Pathria and P.D. Beale, Statistical Mechanics, Butterworth-Heinemann (2011).
O. Contreras-Vergara, N. Lucero-Azuara, N. Sánchez-Salas and J.I. Jiménez-Aquino, Harmonic oscillator Brownian motion: Langevin approach revisited, Revista Mexicana de Física E 18 (2021) 97.
G. Mandal, P. Nayak and S.R. Wadia, Coadjoint orbit action of Virasoro group and two-dimensional quantum gravity dual to SYK/tensor models, JHEP 11 (2017) 046 [arXiv:1702.04266] [INSPIRE].
M. Banados, Three-dimensional quantum geometry and black holes, AIP Conf. Proc. 484 (1999) 147 [hep-th/9901148] [INSPIRE].
Acknowledgments
We would like to thank Soumyadeep Chaudhuri, R Loganayagam, Juan Maldacena, Shiraz Minwalla, Kyriakos Papadodimas, Onkar Parrikar, Suvrat Raju, Subir Sachdev, Ashoke Sen, Ritam Sinha, Nilakash Sorokhaibam, Sandip Trivedi and Neha Wadia for discussions and comments during the course of this work. S.R.W. would like to thank the support of the Infosys Foundation Homi Bhabha Chair at ICTS-TIFR. A.K. and G.M. acknowledge support from the Quantum Space-Time Endowment of the Infosys Science Foundation.
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Gaikwad, A., Kaushal, A., Mandal, G. et al. A microscopic model of black hole evaporation in two dimensions. J. High Energ. Phys. 2023, 171 (2023). https://doi.org/10.1007/JHEP08(2023)171
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DOI: https://doi.org/10.1007/JHEP08(2023)171